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EXISTENCE OF ALMOST AUTOMORPHIC
SOLUTION IN DISTRIBUTION FOR A CLASS OF
STOCHASTIC INTEGRO-DIFFERENTIAL
EQUATION DRIVEN BY LÉVY NOISE
Mamadou Moustapha Mbaye, Solym Manou-Abi
To cite this version:
arXiv:1911.07811v1 [math.PR] 18 Nov 2019
FOR A CLASS OF STOCHASTIC INTEGRO-DIFFERENTIAL EQUATION DRIVEN BY L ´EVY NOISE
MAMADOU MOUSTAPHA MBAYE AND SOLYM MAWAKI MANOU-ABI
Abstract. We investigate a class of stochastic integro-differential equations driven by L´evy noise. Under some appropriate assumptions, we establish the existence of an square-mean almost automorphic solutions in distribution. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution distribution is obtained by using the condition which is weaker than Lipschitz conditions. We provide an example to illustrate ours results.
1. Introduction
The aim of this work is to study the existence and uniqueness of the square-mean al-most automorphic mild solutions in distibution to the following class of nonlinear stochas-tic integro-differential equations driven by L´evy noise in a separable Hilbert space H
x′(t) = Ax(t) + g(t, x(t)) + Z t −∞ B1(t− s) f (s, x(s))ds (1.1) + Z t −∞ B2(t− s)h(s, x(s))dW(s) + Z t −∞ B2(t− s) Z |y|V<1 F(s, x((s−), y) ˜N(ds, dy) + Z t −∞ B2(t− s) Z |y|V≥1
G(s, x(s−, y)N(ds, dy) for all t ∈ R,
where A : D(A)⊂ H is the infinitesimal generator of a C0-semigroup (T (t))t≥0, B1and B2
are convolution-type kernels in L1(0,∞) and L2(0,∞) respectively. g, f : R × L2(P, H)→ L2(P, H) h : R× L2(P, H) → L(V, L2(P, H)) F, G : R× L2(P, H)× V → L2(P, H); W and N are the L´evy-Itˆo decomposition components of the two-sided L´evy process L (with assumptions stated in Section2.).
Throughout this work, we assume (H,k · k) and (V, | · |) are real separable Hilbert spaces. We denote by L(V, H) the family of bounded linear operators from V to H and L2(P, H) is the space of all H-valued random variables x such that
Ekxk2= Z
Ωkxk
2dP < +
∞.
2000 Mathematics Subject Classification. 34C27; 34K14; 34K30; 34K50; 35B15; 35K55; 43A60; 60G20. Key words and phrases. almost automorphic solution; stochastic processes; stochastic evolution equations; L´evy noise.
The concept of almost automorphic is a natural generalization of the almost periodicity that was introduced by Bochner [6]. The basic aspects of the theory of almost automorphic functions can be found for instance in to the book [21].
In recent years, the study of almost periodic or almost automorphic solutions to some stochastic differential equations have been considerably investigated in lots of publications [2,3,4,5,7,8,9,10,11,13,24] because of its significance and applications in physics, mechanics and mathematical biology. The concept of square-mean almost automorphic stochastic processes was introduced by Fu and Liu [14]. As indicated in [15,16], it ap-pears that almost periodicity or automorphy in distribution sense is a more appropriate concept relatively to solutions of stochastic differential equations. Recently, the concept of Poisson square-mean almost automorphy was introduced by Liu and Sun [17] to deal with some stochastic evolution equations driven by L´evy noise. For the almost automorphy in distribution, its various extensions in distribution sense and the applications in stochastic differential equations, one can see [12,19,26] for more details.
One should point out that other slightly different versions of equation (1.1) have been considered in the literature. In particular, Bezandry [3], Xia [25] and Mbaye [18] inves-tigated the existence and uniqueness of the solution of equation (1.1) in the case when F = G =0 based on anotherl fixed point theorem.
The rest of this work is organized as follows. In Section 2, we make a recalling on L´evy process. In Section 3, we review some concepts and basic properties on almost automorphic and Poisson almost automorphic processes. In Section 4, by Schauder’s fixed point theorem, we prove the existence of an square-mean almost automorphic mild solution in distribution of equation (1.1). In Section 5, we provide an example to illustrate our results.
2. L´evy process
Definition 2.1. A V-valued stochastic process L = (L(t), t ≥ 0) is called L´evy process if: (1) L(0) = 0 almost surely;
(2) L has independent and stationary increments;
(3) L is stochastically continuous, i.e. for all ǫ > 0 and for all s > 0 lim
t→sP(|L(t) − L(s)| > ǫ) = 0.
Every L´evy process is c`adl`ag. Let L be a L´evy process. Definition 2.2. [17]
(1) A borel B in V− {0} is bounded below if 0 < B, where B is the closure of B. (2) ν(·) = E(N(1, ·)) is called the intensity measure associated with L, where △L(t) =
L(t)− L(t−) for each t ≥ 0 and N(t, B)(ω) := ♯ 0≤ s ≤ t : △L(s)(ω) ∈ B := X 0≤s≤t χB(△L(s)(ω))
with L(t−) = limtրsL(s) and χB being the indicator function for any Borel set B
in V− {0}.
(3) N(t, B) is called Poisson random measure if B is bounded below, for each t≥ 0. (4) For each t≥ 0 and B bounded below, we define the compensated Poisson random
measure by
e
Proposition 2.1. [1,22] Let L be the V-valued L´evy process. Then there exist a∈ V, V-valued Wiener process W with covariance operator Q, and an independent Poisson random measure on R+× (V − {0}) such that for each t ≥ 0
L(t) = at + W(t) + Z |x|<1 x eN(t, dx) + Z |x|≥1 xN(t, dx), where the Poisson random measure N has the intensity measure ν satisfying
(2.1)
Z
V
(|x|2∧ 1)ν(dx) < ∞ and eN is the compensated Poisson random measure of N.
Let L1(t) and L2(t), t≥ 0 be two independent and identically distributed L´evy processes.
Let
L(t) = (
L1(t) for t≥ 0,
−L2(−t) for t <0
Remark 2.1. By (2.1), it follows thatR
|x|≥1ν(dx) < ∞. For convenience, we denote
b:= Z
|x|≥1
ν(dx).
Then L is a two-sided L´evy process defined on the filtered probability space (Ω,F , P, (Ft)t∈R).
We assume that Q is a positive, self-adjoint and trace class operator on V, see [23] for more details. The stochastic process eL =(eL(t), t∈ R) given by eL(t) := L(t + s)− L(s) for some s∈ R is also a two-sided L´evy process which shares the same law as L. For more details about the L´evy process, we refer to [1,17,22].
3. Square-mean almost automorphic process
In this section, we recall the concepts of square-mean almost automorphic process and there basic properties.
Definition 3.1. Let x : R → L2(P, H) be a stochastic process.
(1) x is said to be stochastically bounded if there exists M > 0 such that Ekx(t)k2 ≤ M for all t ∈ R.
(2) x is said to be stochastically continuous if lim
t→s
Ekx(t) − x(s)k2=0 for all s∈ R.
Denote by S BC(R, L2(P, H)) the space of all the stochastically bounded and
continu-ous processes. Clearly, the space S BC(R, L2(P, H)) is a Banach space equipped with the
following norm
kxk∞=sup
t∈R
(Ekx(t)k2
)12.
Definition 3.2. [17] Let J : R× V → L2(P, H) be a stochastic process.
(1) J is said to be Poisson stochastically bounded if there exists M > 0 such that Z
V
(2) J is said to be Poisson stochastically continuous if lim t→s Z V EkJ(t, x) − J(s, x)k2ν(dx) = 0 for all s∈ R.
Denote by PS BC(R× V, L2(P, H)) the space of all the stochastically bounded and
continuous processes. Definition 3.3. [14] Let x : R→ L2
(P, H) be a continuous stochastic process. x is said be square-mean almost automorphic process if for every sequence of real numbers (t′n)n we
can extract a subsequence (tn)nsuch that, for some stochastic process y : R → L2(P, H),
we have
lim
n→+∞
Ekx(t + tn)− y(t)k2=0 for all t∈ R and
lim
n→+∞
Eky(t − tn)− x(t)k2=0 for all t∈ R. We denote the space off all such stochastic processes by S AA(R, L2(P, H)).
Theorem 3.1. [14] S AA(R, L2(P, H)) equipped with the normk · k∞is a Banach space.
Definition 3.4. [17] Let D : R× V → L2(P, H) be stochastic process. D is said be Poisson
square-mean almost automorphic process in t∈ R if D is Poisson continuous and for every sequence of real numbers (tn′)n we can extract a subsequence (tn)n such that, for some
stochastic process ˜D: R× V → L2
(P, H) withRVEk ˜D(t, x)k2ν(dx) < ∞ such that lim n→+∞ Z V EkD(t + tn, x) − ˜D(t, x)k2ν(dx) = 0 for all t∈ R and lim n→+∞ Z V Ek ˜D(t− tn, x) − D(t, x)k2ν(dx) = 0 for all t∈ R. We denote the space off all such stochastic processes by PS AA(R× V, L2(P, H)).
Definition 3.5. [17] Let F : R× L2(P, H)
× V → L2(P, H) be stochastic process. F is
said be Poisson square-mean almost automorphic process in t ∈ R for each Y ∈ L2(P, H)
if F is Poisson continuous and for every sequence of real numbers (t′n)nwe can extract a
subsequence (tn)nsuch that, for some stochastic process ˜F : R× L2(P, H)× V → L2(P, H)
withRVEk ˜F(t, Y, x)k2ν(dx) < ∞ such that lim n→+∞ Z V EkF(t + tn, Y, x) − ˜F(t, Y, x)k2ν(dx) = 0 for all t∈ R and lim n→+∞ Z V Ek ˜F(t− tn, Y, x) − F(t, Y, x)k2ν(dx) = 0 for all t∈ R. We denote the space off all such stochastic processes by PS AA(R× L2
(P, H)× V, L2
(P, H)). LetP(H) be the space of all Borel probability measures on H with the β metric.
β(µ, ν) := sup{| Z
f dµ− Z
f dν| : k f kBL≤ 1}, µ, ν ∈ P(H),
where f are Lipschitz continuous real-valued functions on H with
k f kBL=k f kL+k f k∞, k f kL=sup x,y
| f (x) − f (y)|
Definition 3.6. [17] An H-valued stochastic process Y(t) is said to be almost automorphic in distribution if its law µ(t) is aP(H)-valued almost automorphic mapping, i.e. for every sequence of real numbers (s′n)n, there exist a subsequence (sn)nand aP(H)-valued mapping
˜µ(t) such that lim
n→∞β(µ(t + sn), ˜µ(t)) = 0 and nlim→∞β( ˜µ(t − sn), µ(t)) = 0
hold for each t∈ R.
To study the existence of mild solutions to the stochastic evolution equations (1.1). we will need the following assumptions,
(H.1) The semigroup T (t) is compact for t > 0 and is exponentially stable, i.e., there exists constants K, ω > 0 such that
(3.1) kT (t)k ≤ Ke−ωt for all t≥ 0.
(H.2) The functions f , g and h are uniformly continuous on any bounded subset K of L2(Ω, H) for each t
∈ R. F, G are uniformly continuous on any bounded subset K of L2(Ω, H) for each t ∈ R and x ∈ V. For each bounded subset K ⊂ L2(Ω, H), g(R, K), f (R, K), h(R, K) are bounded and F(R, K, V) and G(R, K, V) are Poisson stochastically bounded. Moreover we suppose that there exists r > 0 such that (3.2) ∆r≤ ω2r 20θK2 where ∆r=max ( sup t∈RkukL2≤r f(t, u) L2, sup t∈RkukL2≤r g(t, u) L2, sup t∈RkukL2≤r h(t, u) L2, sup t∈RkukL2≤r Z |y|V<1 F(t, u, x) L2ν(dx), sup t∈RkukL2≤r Z |y|V≥1 G(t, u, x) L2ν(dx) ) and θ = max1,kB1k2L1(0,∞), 4kB2k2L2(0,∞), 2bkB2k2L1(0,∞)
(H.3) we suppose that there exist measurable functions mg, mf mh, mF, mG : R −→
[0,∞) such that (3.3) Ek g(t, Y) − g(t, Z) k2≤ mg(t)· E k Y − Z k2 (3.4) Ek f (t, Y) − f (t, Z) k2≤ mf(t)· E k Y − Z k2 (3.5) Ek (h(t, Y) − h(t, Z))Q12 k2 L(V,L2(P,H))≤ mh(t)· E k Y − Z k2 (3.6) Z |x|V<1 EkF(t, Y, x) − F(t, Z, x)k2ν(dx) ≤ mF(t)· EkY − Zk2 (3.7) Z |x|V≥1 EkG(t, Y, x) − G(t, Z, x)k2ν(dx) ≤ mG(t)· EkY − Zk2 for all t∈ R and for any Y, Z ∈ L2(P, H).
(H.4) If (un)n∈IN ⊂ S BC(R, L2(P, H)) is uniformly bounded and uniformly convergent
upon every compact subset of R, then g(·, un(·)), f (·, un(·)), h(·, un(·)), and F(·, un(·), ·),
G(·, un(·), ·) are relatively compact in S BC(R, L2(P, H)), PS BC(R× V, L2(P, H)),
Definition 3.7. An Ft-progressively measurable process{x(t)}t∈Ris called a mild solution
on R of equation (1.1) if it satisfies the corresponding stochastic integral equation
x(t) = T(t− a)x(a) + Z t a T(t− s)g(s, x(s))ds (3.8) + Z t a T(t− σ) Z σ a B1(σ− s) f (s, x(s))dsdσ + Z t a T(t− σ) Z σ a B2(σ− s)h(s, x(s))dW(s)dσ + Z t a T(t− σ) Z σ a B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t a T(t− σ) Z σ a B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ for all t≥ a.
Remark 3.1. If we let a → −∞ in the stochastic integral equation (3.8), by the exponential dissipation condition of (T (t))t≥0, then we obtain the stochastic process x : R→ L2(Ω, H)
is a mild solution of the equation (3.8) if and only if x satisfies the stochastic integral equation x(t) = Z t −∞ T(t− s)g(s, x(s))ds + Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ (3.9) + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ.
4. Square-mean almost automorphic solutions
This section is devoted to the existence and the uniqueness of the square-mean almost automorphic mild solution in distribution on R of Eq. (1.1).
Lemma 4.1. If the semigroup T (t) verifies (3.1) in assumption(H.1) and if the functions f , g and h are uniformly continuous on any bounded subset K of L2(Ω, H) for each t∈ R. F, G are uniformly continuous on any bounded subset K of L2(Ω, H) for each t ∈ R and x ∈ V. For each bounded subset K ⊂ L2(Ω, H), g(R, K), f (R, K), h(R, K) are bounded and F(R, K, V) and G(R, K, V) are Poisson stochastically bounded., then the mapping Λ : S BC(R, L2(P, H))
→ S BC(R, L2(P, H)) is well-defined and continuous.
Proof. It is easy to see that S is well-defined. To complete the proof it remains to show that Λis continuous. Consider an arbitrary sequence of functions un ∈ S BC(R, L2(P, H)) that
converges uniformly to some u∈ S BC(R, L2
(P, H)), that is, un− u
∞ → 0 as n → ∞.
There exists a bounded subset K of L2(Ω, H) such that un(t), u(t) ∈ K for each t ∈ R
and n = 1, 2, .... By assumptions, given ǫ > 0, there exist δ > 0 and N > 0 such that Ekun(t)− u(t)k2< δ imply that
≤ 5E Z t −∞ T(t− s) g(s, un(s))− g(s, u(s)) ds 2 +5E Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f(s, un(s))− f (s, u(s)) dsdσ 2 +5E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) h(s, un(s))− h(s, u(s)) dW(s)dσ 2 +5E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, un(s−), y) − F(s, u(s−), y)) ˜ N(ds, dy)dσ 2 +5E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, un(s−), y) − G(s, u(s−), y)) N(ds, dy)dσ 2 ≤ 5(I1+ I2+ I3+ I4+ I5).
Using Cauchy-Schwartz’s inequality, we get
I1<
ǫ
25 I2< ǫ 25
For I3, using Cauchy-Schwartz’s inequality and Ito’s isometry property, we obtain
I3≤ K2 ω Z t −∞ e−ω(t−σ) Z σ −∞ E B2(σ− s) h(s, un(s))− h(s, u(s)) Q12 2 dsdσ < ǫ 25
As to I4and I5, by Cauchy-Schwartz’s inequality and the properties of the integral for the
Poisson random measure, we have
And I5 = E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 F(s, un(s−), y) − F(s, u(s−), y)) N(ds, dy)dσ 2 ≤ 2E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 F(s, un(s−), y) − F(s, u(s−), y)) ˜ N(ds, dy)dσ 2 +2E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 F(s, un(s−), y) − F(s, u(s−), y)) ν(dy)dσ 2 ≤2K 2 ω2 kB2k 2 L2(0,∞) ω2ǫ 50K2(kB 2k2L2(0,∞)+ bkB2k 2 L1(0,∞)+1) ! +2K 2 ω Z t −∞ e−ω(t−σ)E Z σ −∞ B2(σ− s) Z |y|V≥1 F(s, un(s−), y) − F(s, u(s−), y)) ν(dy)ds 2 dσ ≤2K 2 ω2 kB2k 2 L2(0,∞) ω2ǫ 50K2(kB 2k2L2(0,∞)+ bkB2k2L1(0,∞)+1) ! +2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞ B2(σ− s) ds Z |y|V≥1 ν(dy) · Z σ −∞ B2(σ− s) Z |y|V≥1 E F(s, un(s−), y) − F(s, u(s−), y) 2 ν(dy)dsdσ ≤2K 2 ω2 kB2k 2 L2(0,∞) ω2ǫ 50K2(kB 2k2L2(0,∞)+ bkB2k 2 L1(0,∞)+1) ! +2bK 2 ωkB2k 2 L1(0,∞) ω2ǫ 50K2(kB 2k2L2(0,∞)+ bkB2k 2 L1(0,∞)+1) ! < ǫ 25.
Thus, by combining I1− I5, it follows that for each t∈ R and n > N
Ek(Λun)(t)− (Λu)(t)k2< ǫ.
This implies that Λ is continuous. The proof is complete. Theorem 4.1. Assume that assumptions (H.1) − (H.4) hold and
1- g, f ∈ S AA(R × L2(P, H), L2(P, H)), 2- h∈ S AA(R × L2(P, H), L(V, L2(P, H))), 3- F∈ PS AA(R × L2(P, H) × V, L2(P, H)) 4- G∈ PS AA(R × L2(P, H) × V, L2(P, H)).
Proof. Let B = {u ∈ S BC(R, L2
(P, H)) : kuk2
∞ ≤ r}. By Lemma4.1and (3.2), it follows
that B is a convex and closed set satisfying ΛB ⊂ B. To complete the proof, we have to prove the following statements:
a) That V ={Λu(t) : u ∈ B} is a relatively compact subset of L2
(P, H) for each t∈ R; b) That U ={Λu : u ∈ B} ⊂ S BC(R, L2(P, H) is equi-continuous.
c) The mild solution is almost automorphic in distribution.
To prove a), fix t∈ R and consider an arbitrary ε > 0. Then
(Λǫx)(t) = Z t−ǫ −∞ T(t− s)g(s, x(s))ds + Z t−ǫ −∞ T(t− σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ + Z t−ǫ −∞ T(t− σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ + Z t−ǫ −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t−ǫ −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ = T(ǫ) Z t−ǫ −∞ T(t− ǫ − s)g(s, x(s))ds + Z t−ǫ −∞ T(t− ǫ − σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ + Z t−ǫ −∞ T(t− ǫ − σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ + Z t−ǫ −∞ T(t− ǫ − σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t−ǫ −∞ T(t− ǫ − σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ = T(ǫ)(Λx)(t− ǫ)
and hence Vε := {(Λε)x(t) : x ∈ B} is relatively compact in L2(P, H) as the semigroup
family T (ε) is compact by assumption. Now Ek(Λu)(t) − (Λǫu)(t)k2= E Z t t−ǫ T(t− s)g(s, u(s))ds + Z t t−ǫ T(t− σ) Z σ −∞ B1(σ− s) f (s, u(s))dsdσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) h(s, un(s))− h(s, u(s)dW(s)dσ + Z t t−ǫ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, u(s−), y) ˜N(ds, dy)dσ + Z t t−ǫ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1
G(s, u(s−), y)N(ds, dy)dσ.
≤ 5E Z t t−ǫ T(t− s)g(s, u(s))ds 2 +5E Z t t−ǫ T(t− σ) Z σ −∞ B1(σ− s) f (s, u(s))dsdσ 2 +5E Z t t−ǫ T(t− σ) Z σ −∞ B2(σ− s)h(s, u(s))dW(s)dσ 2 +5E Z t t−ǫ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, u(s−), y) ˜N(ds, dy)dσ 2 +5E Z t t−ǫ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1
G(s, u(s−), y)N(ds, dy)dσ
2 ≤ 5∆rK2 Z t t−ǫ e−ω(t−σ)dσ 2 + ∆rK2kB1k2L1(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + ∆rK2kB2k2L2(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + ∆rK2kB2k2L2(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + 2∆rK2kB2k2L2(0,∞)+2b∆rK2kB2k2L1(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 ≤ 5∆rK2 1 +kB1k2L1(0,∞)+4kB2k2L2(0,∞)+2bkB2k2L1(0,∞) ǫ2
from which it follows that V ={Λu(t) : u ∈ B} is a relatively compact subset of L2
(P, H) for each t∈ R.
We now show that b) holds. Let u ∈ B and t1, t2 ∈ R such that t1 < t2. Similar
computation as that in Lemma4.1, we have
= E T(t2− t1)− I Z t1 −∞ T(t1− s)g(s, x(s))ds + Z t1 −∞ T(t1− σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ + Z t1 −∞ T(t1− σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ + Z t1 −∞ T(t1− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t1 −∞ T(t1− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ + Z t2 t1 T(t2− s)g(s, x(s))ds + Z t2 t1 T(t2− σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ + Z t2 t1 T(t2− σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ + Z t2 t1 T(t2− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ + Z t2 t1 T(2− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ 2 ≤ 6 sup y∈V E T(t2− t1)− I y 2 +6E Z t2 t1 T(t2− s)g(s, x(s))ds 2 + 6E Z t2 t1 T(t2− σ) Z σ −∞ B1(σ− s) f (s, x(s))dsdσ 2 + 6E Z t2 t1 T(t2− σ) Z σ −∞ B2(σ− s)h(s, x(s))dW(s)dσ 2 + 6E Z t2 t1 T(t2− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s, x(s−), y) ˜N(ds, dy)dσ 2 + 6E Z t2 t1 T(2− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s, x(s−), y)N(ds, dy)dσ 2 ≤ 6 sup y∈V E T(t2− t1)− I y 2 +6 ∆rK2 Z t2 t1 e−ω(t2−σ)dσ 2 + ∆rK2kB1k2L1(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + ∆rK2kB2k2L2(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + ∆rK2kB2k2L2(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 + 2∆rK2kB2k2L2(0,∞)+2b∆rK2kB2k2L1(0,∞) Z t t−ǫ e−ω(t−σ)dσ 2 ≤ 6 sup y∈V E T(t2− t1)− I y 2 + 6∆rK2 1 +kB1k2L1(0,∞)+4kB2k2L2(0,∞)+2bkB2k2L1(0,∞) Z t2 t1 e−ω(t2−σ)dσ 2 .
The right-hand side tends to 0 independently to u∈ B as t2 → t1which implies that U is
Denote the closed convex hull of ΛB by co ΛB. Since co ΛB ⊂ B and B is a closed convex, it follows that
Λ(co ΛB)⊂ ΛB ⊂ co ΛB.
Further, it is not hard to see that co ΛB is relatively compact in L2(P, H). Using
Arzel`a-Ascoli theorem, we deduce that the restriction of co ΛB to any compact subset I of R is relatively compact in C(I, L2(P, H)). Thus condition (H.4) implies that Λ : co ΛB
7→ co ΛB is a compact operator. In summary, S : co ΛB7→ co ΛB is continuous and compact. Using the Schauder fixed point it follows that Λ has a fixed-point.
To end the proof, we have to check this the fixed-point is almost automorphic in distri-bution. Since g, f , h are almost automorphic and F, G are Poisson almost automorphic, then for every sequence of real numbers (t′n)nwe can extract a subsequence (tn)nsuch that,
for some stochastic processeseg, ef, eh, eF, eG (4.2) lim n→+∞ Ekg(s + tn, X) − eg(s)k2=0, lim n→+∞ Ekeg(s− tn, X) − g(s, X)k2=0; (4.3) lim n→+∞ Ek f (s + tn, X) − ef(s, X)k2=0, lim n→+∞ Ek ef(s− tn, X) − f (s, X)k2=0; (4.4) lim n→+∞ Ek h(s+tn, X)−eh(s, X) Q12k2 L(V,L2(P,H))=0, lim n→+∞ Ek eh(s−tn, X)−h(s, X) Q12k2 L(V,L2(P,H))=0; lim n→+∞ Z |y|V<1 EkF(s + tn, X, y) − eF(s, X, y)k2ν(dy) = 0, (4.5) lim n→+∞ Z |y|V<1 EkeF(s− tn, X, y) − F(s, X, y)k2ν(dy) = 0 and lim n→+∞ Z |y|V≥1 EkG(s + tn, X, y) − eG(s, X, y)k2ν(dy) = 0, (4.6) lim n→+∞ Z |y|V≥1 Ek eG(s− tn, X, y) − G(s, X, y)k2ν(dy) = 0 hold for each s∈ R and X ∈ L2
(P, H). For t∈ R, we define e X(t) = Z t −∞ T(t− s)eg(s, eX(s))ds + Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) ef(s, eX(s))dsdσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s)eh(s, eX(s))dW(s)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 e F(s, eX(s−), y) ˜N(ds, dy)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 e G(s, eX(s−), y)N(ds, dy)dσ. Let Wn(s) = W(s + tn)− W(tn), Nn(s, y) = N(s + tn, y) − N(tn, x) and ˜Nn(s, y) = N(s + tn, y) −
N(tn, x) for each s ∈ R. Then Wnis also a Q-Wiener process having the same distribution
as W and Nn have the same distribution as N with compensated Poisson random measure
Consider the process define as follows Xn(t) = Z t −∞ T(t− s)g(s + tn, Xn(s))ds + Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f (s + tn, Xn(s))dsdσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s)h(s + tn, Xn(s))dW(s)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V<1 F(s + tn, Xn(s−), y) ˜N(ds, dy)dσ + Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) Z |y|V≥1 G(s + tn, Xn(s−), y)N(ds, dy)dσ.
Note that X(t + tn) and Xn have the same law and since the convergence in L2 implies
For J2, we have J2= E Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f(s + tn, Xn(s))− ef(s, eX(s)) dsdσ 2 ≤ 2E Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f(s + tn, Xn(s))− f (s + tn, eX(s)) dsdσ 2 +2E Z t −∞ T(t− σ) Z σ −∞ B1(σ− s) f(s + tn, eX(s))− ef(s, eX(s)) dsdσ 2 ≤2K 2 ω kB1kL1(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 1(σ− s)kmf(s + tn)E Xn(s)− eX(s) 2 dsdσ +2K 2 ω kB1kL1(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 1(σ− s)kEk f (s + tn, eX(s))− ef(s, eX(s))k2dsdσ ≤2K 2 ω kB1kL1(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 1(σ− s)kmf(s + tn)E Xn(s)− eX(s) 2 dsdσ + ζ2n, where ζn 2 := 2K2
ω2kB1k2L1(0,∞)sups∈REk f (s + tn, eX(s))− ef(s, eX(s))k2. For the same reason as
for ζn
1, ζ
n
2 → 0 as n → ∞.
For J3, using Cauchy-Schwartz’s inequality and the Ito’s isometry, we have
J3 = E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) h(s + tn, Xn(s))− eh(s, eX(s)) dW(s)dσ 2 ≤ 2E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) h(s + tn, Xn(s))− h(s + tn, eX(s)) dW(s)dσ 2 +2E Z t −∞ T(t− σ) Z σ −∞ B2(σ− s) h(s + tn, eX(s))− eh(s, eX(s)) dW(s)dσ 2 ≤2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mh(s + tn)E Xn(s)− eX(s) 2 dsdσ +2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2E h(s + tn, eX(s))− eh(s, eX(s)) Q12 2 L(V,L2(P,H)) dsdσ ≤2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mh(s + tn)E Xn(s)− eX(s) 2 dsdσ + ζ3n, where ζn 3 := 2K2 ω2kB2k2L2(0,∞)sups∈RE h(s + tn, eX(s))− eh(s, eX(s)) Q12 2 L(V,L2(P,H)) . By(4.4), it follows that ζn 3 → 0 as n → ∞, like ζ n 1.
For J4, using Cauchy-Schwartz’s inequality and the properties of the integral for the
Poisson random measure, we have
≤ 2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mF(s + tn)E Xn(s)− eX(s) 2 dsdσ +2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2 Z |y|V<1 E F(s+ tn, eX(s−), y) − eF(s, eX(s−), y) 2 ν(dy)dsdσ ≤ 2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mF(s + tn)E Xn(s)− eX(s) 2 dsdσ + ζ4n, where ζ4n := 2Kω22kB2k2L2(0,∞)sups∈R R |y|V<1 E F(s + tn, eX(s−), y) − eF(s, eX(s−), y) 2 ν(dy). Using (4.5), it follows that ζn
4 → 0 as n → ∞, like ζ n
1.
For J5, using Cauchy-Schwartz’s inequality and the properties of the integral for the
Poisson random measure, we have
≤ 4K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mG(s + tn)E Xn(s)− eX(s) 2 dsdσ +4bK 2 ω kB2kL1(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)kmG(s + tn)E Xn(s)− eX(s) 2 dsdσ +2K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2 Z |y|V≥1 E G(s+ tn, eX(s−), y) − eG(s, eX(s−), y) 2 ν(dy)dsdσ +4bK 2 ω kB2kL2(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k Z |y|V≥1 E G(s+ tn, eX(s−), y) − eG(s, eX(s−), y) 2 ν(dy)dsdσ ≤ 4K 2 ω Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)k2mG(s + tn)E Xn(s)− eX(s) 2 dsdσ +4bK 2 ω kB2kL1(0,∞) Z t −∞ e−ω(t−σ) Z σ −∞kB 2(σ− s)kmG(s + tn)E Xn(s)− eX(s) 2 dsdσ + ζ5n(t) where ζn 5 = 2K2 ω2 kB2k 2 L2(0,∞)sup s∈R Z |y|V≥1 E G(s+ tn, eX(s−), y) − eG(s, eX(s−), y) 2 ν(dy) +4bK 2 ω2 kB2k 2 L1(0,∞)sup s∈R Z |y|V≥1 E G(s+ tn, eX(s−), y) − eG(s, eX(s−), y) 2 ν(dy) Using (4.6), it follows that ζn
5 → 0 as n → ∞, like ζ n
1. By combining the estimations
J1− J5, we get for all t∈ R
E Xn(t)− eX(t) 2 ≤ ζn+ϑ sup s∈R E Xn(s)− eX(s) 2 where ζn = Σ5 i=1ζ n i. It follows that E Xn(t)− eX(t) 2 ≤ ζ n 1− ϑ. Since ϑ < 1 and ζn
→ 0 as n → ∞ for all t ∈ R then one has E Xn(t)− eX(t) 2 → 0 as n → 0 for each t ∈ R.
Hence we deduce that X(t + tn) converge to eX(t) in distribution. Similarly, one can get
that eX(t− tn) converge to X(t) in distribution too. Therefore this fixed-point solution of
the equation (1.1) is square-mean almost automorphic in distribution. which completes the proof.
5. Example
where W is a Q-Wiener process on L2(0, 1) with Tr Q < ∞ and Z is a L´evy pure jump process on L2(0, 1) which is independent of W and ω > 0. The forcing terms are follows:
g(t, u) = δ sin u(sin t + sin√2t), f(t, u) = δ sin u(sin t + sin√3t),
h(t, u) = δ sin u(sin t + sin√5t), θ(t, u) = δ sin u(sin t + sin πt)
with δ > 0. Denote H = V = L2(0, 1). In order to write the system (5.1) on the abstract
form (1.1), we consider the linear operator A : D(A)⊂ L2(0, 1)
→ L2(0, 1), given by
D(A) = H2(0, 1)∩ H01(0, 1),
Ax(ξ) = x”(ξ) for ξ ∈ (0, 1) and x ∈ D(A).
It is well-known that A generates a C0-semigroup (T (t))t≥0on L2(0, 1) defined by
(T (t)x)(r) = ∞ X n=1 e−n2π2thx, eniL2en(r), where en(r) = √
2 sin(nπr) for n = 1, 2, ...., andkT (t)k ≤ e−π2tfor all t≥ 0. Then the system (5.1) takes the following abstract form
u′(t) = Au(t) + g(t, u(t)) + Z t −∞ B1(t− s) f (s, u(s))ds + Z t −∞ B2(t− s)h(s, u(s))dW(s) + Z t −∞ B2(t− s) Z |y|V<1 F(s, u((s−), y) ˜N(ds, dy) + Z t −∞ B2(t− s) Z |y|V≥1
G(s, u(s−, y)N(ds, dy) for all t ∈ R, where u(t) = u(t,·), B1(t) = B2(t) = eωtfor t≥ 0 and
θ(t, u)Z(dt) = Z |y|V<1 F(t, u(t−), y) ˜N(dt, dy) + Z |y|V≥1
G(t, u(t−), y)N(dt, dy) with Z(t) = Z |y|V<1 y ˜N(t, dy) + Z |y|V≥1 yN(t, dy),
F(t, u, y) = h(t, u)y· 1{|y|V<1} and G(t, u, y) = h(t, u)y· 1{|y|V≥1}.
Here, we assume that the L´evy pure jump process Z on L2(0, 1) is decomposed as above by the L´evy-It decomposition theorem.
Clearly, f , θ are almost automorphic, and F, G Poisson almost automorphic and satisfying H1–H3. Moreover, it is easy to see that the conditions (3.3)–(3.7) are satisfied with
Obviously, Lg=sup t∈R Z t −∞ e−ω(t−s)mg(s)ds = δ2sup t∈R Z t −∞ e−ω(t−s)mg(s)ds sin s + sin√2s 2 < ∞, Lf =sup t∈R Z t −∞kB 1(t− s)kmf(s)ds = δ2sup t∈R Z t −∞kB 1(t− s)k sin s + sin√3s 2 ds <∞, Lh=sup t∈R Z t −∞kB 2(t−s)kmh(s)ds = δ2kQkL(V,V)sup t∈R Z t −∞kB 2(t−s) sin s+sin√5s 2 ds <∞, LF =sup t∈R Z t −∞kB 2(t−s)k2mF(s)ds = δ2ν(B1(0)) sup t∈R Z t −∞kB 2(t−s)k2 sin s+sin πs 2 ds <∞, LG=sup t∈R Z t −∞kB 2(t− s)k2mG(s)ds = δ2bsup t∈R Z t −∞kB 2(t− s)k2 sin s + sin πs 2 ds <∞ Therefore, by Theorem4.1, the equation (5.1) has a square-mean almost automorphic in distribution mild solution on R whenever δ is small enough.
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D´epartement de Math´ematiques, Facult´e des Sciences et Technique, Universit´e Cheikh Anta Diop, BP 5005, Dakar-Fann, Senegal
E-mail address: mamadoumoustapha3.mbaye@ucad.edu.sn
Institut Montpelli´erain Alexander Grothendieck, UMR CNRS 5149, Universit´e de Montpellier E-mail address: solym-mawaki.manou-abi@umontpellier.fr
